Wyniki wyszukiwania - BazTech

1

Laplace transform for fractional differential equations in viscoelastic electrical systems

Bouzidi Mohammed, Nasri Abdelfatah, Hamouda Messaoud

Diagnostyka

|

2025

|

Vol. 26, No. 3

art. no. 2025305

EN

This paper studies the use of the Laplace transform as a key tool for solving fractional differential equations which involve non-integer derivatives and are used to model various physical phenomena such as viscoelastic materials and control systems Fractional differential equations pose significant challenges due to the complexity of fractional derivatives and integral terms making classical solution methods inefficient The methodology in the paper relies on the Laplace transform to convert fractional equations shifting to a frequency domain to the temporal domain simplifying the handling of these complex equations This approach enables precise and efficient solutions and transforms complex equations into more manageable forms The study also explores practical applications such as solving equations related to viscoelastic materials which exhibit dynamic behavior governed by fractional equations This contributes to a deeper understanding of these materials and their mathematical modeling The paper concludes that the Laplace transform offers a robust framework for solving a wide range of fractional differential equations more efficiently with significant benefits in mathematical modeling and analysis Additionally the study highlights the importance of integrating digital methods with the Laplace transform for solving complex boundary problems thereby enhancing practical applications in fields like applied mathematics and engineering.

2

Analytical solution of Atangana-Baleanu fractional viscoelastic relaxation model – Laplacian approach

Khan Muhammad Sabeel, Sagheer Ayesha, Azeem Zarafshan

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2025

|

Vol. 73, nr 5

art. no. e154145

EN

In this contribution, a new novel approach based on the Atangana-Baleanu fractional in conjunction with the Laplacian approach is utilized to obtain an analytical solution of a fractional time relaxation viscoelastic model. The fractional time relaxation model is based on the upper convected Maxwell constitutive relaxation equation. Results for the existence and uniqueness of the solution are presented. Analytical expressions of the solutions are obtained for the underlying physical time relaxation viscoelastic model. Two test model problems with prescribed initial conditions are used to investigate the intricate behaviour of the viscoelastic two-dimensional fluid. The influence of key parameters such as relaxation time, Reynolds number and the order of the fractional derivative on fluid flow characteristics is analyzed and discussed.

3

The velocity field to unsteady fluid flow in a circular cylinder with generalized caputo fractional derivative

Eman Mohamed M., Kader Abass H. Abdel

Journal of Applied Mathematics and Computational Mechanics

|

2024

|

Vol. 23, nr 4

66--75

EN

In this paper, we study the velocity field corresponding to the unsteady flow of a second-grade fluid with a generalized Caputo fractional derivative in a circular cylinder. The analytical solution of the velocity field has been obtained utilizing the Laplace and the finite Hankel transforms. The solution is obtained in terms of a series containing the Mittag-Leffler functions, being the generalization of the exponential function. The effect of the fractional parameters and on fluid motion are illustrated graphically for three different cases. The model discussed in this work is more general and can be applied to other fluid models.

4

On the stability analysis for a semi-analytical scheme for solving the fractional order blood ethanol concentration system using LVIM

Adel Mohamed, Sweilam Nasser H., Khader Mohamed M.

Journal of Applied Mathematics and Computational Mechanics

|

2024

|

Vol. 23, nr 1

7-18

EN

The purpose of this article is to present the Laplace variational iteration method, which combines the VIM with the Laplace transform approach (LVIM). This combination will result in a better and more quickly convergent sequence since nonlinear fractional differential equations (FDEs) cannot be solved using the Laplace transform. With the use of the fixed point theory, the stability analysis is specifically discussed and examined. The blood ethanol concentration system is solved numerically by using the suggested scheme. This model can be represented by a system of FDEs. The investigation will emploi the Caputo-Fabrizio fractional derivative. To provide a more in-depth study of this model, we have taken it in its fractional form so that we can more accurately follow the behavior of the solution in the future and history based on the memory effect of fractional derivatives. We determine the accuracy and efficiency of the provided process by evaluating the absolute errors, and a comparison with the existing published work. The results show that the approach is a useful tool for simulating this model.

5

Inverse laplace transform for a new class of functions and its application to time-fractional diffusion-wave equations

Latif Mohamed S. Abdel, Elhadedy Hager, Kader Abass H. Abdel

Journal of Applied Mathematics and Computational Mechanics

|

2024

|

Vol. 23, nr 4

56--65

EN

This paper investigates the inverse Laplace transform of a certain class of functions. This inverse Laplace transform is obtained in the form of an infinite series of the three-parameter Mittag-Leffler function. Additionally, we found the sum of an infinite series of Mittag-Leffler functions with three parameters in terms of the Wright function. As an application, we get an exact solution of the time-fractional diffusion-wave equation with the Hilfer-Prabhakar time-fractional derivative using Laplace and Fourier transforms.

6

On the Laplace transform

Ramm Alexander G.

Journal of Applied Analysis

|

2024

|

Vol. 30, nr 2

209--213

EN

Sufficient conditions are given for a function F(p) to be the Laplace transform of a function f(t) or a distribution f . No assumption on f is given a priori. It is not even assumed that f = 0 for t < 0.

7

Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function

Yin Hong-Ping, Han Ling-Xiong, Qi Feng

Demonstratio Mathematica

|

2024

|

Vol. 57, nr 1

art. no. 20240041

EN

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

8

On applications of computer algebra systems in queueing theory calculations

Ziółkowski Marcin

Bulletin of the Polish Academy of Sciences. Technical Sciences

|

2024

|

Vol. 72, nr 4

art. no. e150199

EN

In the present paper, the most important aspects of computer algebra systems applications in complicated calculations for classical queueing theory models and their novel modifications are discussed. We mainly present huge computational possibilities of Mathematica environment and effective methods of obtaining symbolic results connected with the most important performance characteristics of queueing systems. First of all, we investigate effective solutions to computational problems appearing in queueing theory such as: finding final probabilities for Markov chains with a huge number of states, calculating derivatives of complicated rational functions of one or many variables with the use of classical and generalized L’Hospital’s rules, obtaining exact formulae of Stieltjes convolutions, calculating chosen integral transforms used often in the above-mentioned theory and possible applications of generalized density function of random variables and vectors in these computations. Some exemplary calculations for practical models belonging both to classical models and their generalizations are attached as well.

9

Semi-analytical scheme with its stability analysis for solving the fractional-order predator-prey equations by using Laplace-VIM

Adel Mohamed, Sweilam Nasser H., Khader Mohamed M.

Journal of Applied Mathematics and Computational Mechanics

|

2023

|

Vol. 22, nr 4

5-17

EN

The article’s goal is to implement a semi-analytical technique named, the Laplace variational iteration method (LVIM), which is the combination of VIM and Laplace transform method. Although both the Laplace transform method and VIM cannot be applied to some nonlinear fractional differential equations (FDEs) individually, this combination will give a fast-convergent solution to the problem under study. The proposed scheme is used to numerically solve a biodynamic system called the Lotka-Volterra system, i.e. Predator-Prey Equations (PPEs). The system of FDEs can be used to represent this scenario, as well as the Caputo-Fabrizio fractional derivative will be used throughout the study. By assessing the residual error function, we can confirm that the given procedure is effective and accurate. The outcomes demonstrate that the technique used is an effective tool for simulating such models.

10

Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃

Inoan Daniela, Marian Daniela

Demonstratio Mathematica

|

2023

|

Vol. 56, nr 1

art. no. 20220198

EN

The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

11

Continuous-state branching processes with migration

Vidmar Matija

Probability and Mathematical Statistics

|

2022

|

Vol. 42, Fasc. 2

227--249

EN

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties of these processes are described. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

12

Exact solution for heat conduction inside a sphere with heat absorption using the regularized Hilfer-Prabhakar derivative

Elhadedy Hager, Latif Mohamed S. Abdel, Nour Hamed M., Kader Abbas H. Abdel

Journal of Applied Mathematics and Computational Mechanics

|

2022

|

Vol. 21, nr 2

27--37

EN

In this article, we utilize the finite Sine-Fourier transform and the Laplace transform for solving fractional partial differential equations with regularized Hilfer-Prabhakar derivative. These transforms are used to get analytical solutions for the time fractional heat conduction equation (TFHCE) with the regularized Hilfer-Prabhakar derivative associated with heat absorption in spherical coordinates. Two cases of Dirichlet boundary conditions are considered by obtaining an analytical solution in each case. The effect of the parameters of the regularized Hilfer-Prabhakar derivative on the heat transfer inside the sphere is discussed using some figures.

13

Approximate analytical solution to the Cattaneo heat conduction model with various laser sources

Nuruddeen Rahmatullah Ibrahim

Journal of Applied Mathematics and Computational Mechanics

|

2022

|

Vol. 21, nr 1

67--78

EN

The present manuscript investigates the role being played by various laser short heating sources in a conduction process of a metallic substrate. The Cattaneo heat conduction model is considered in favour of its finiteness of conduction speed. The analytical solutions for the temperature fields are determined via the application of the Laplace integral transform. Finally, we sought a numerical Laplace inversion scheme where the analytical inversion failed and graphically examined the significance of the heating parameters on the temperature fields.

14

Analysis of deflection in visco-thermoelastic beam resonators subjected to harmonic loading

Chopra Deepti, Singh Prince

International Journal of Applied Mechanics and Engineering

|

2022

|

Vol. 27, no. 2

35--52

EN

This paper analyses the transverse deflection in a homogeneous, isotropic, visco-thermoelastic beam when subjected to harmonic load. The ends of the beam are considered at different boundary conditions (both axial ends clamped, both axial ends simply supported and left end clamped and right end free). The deflection has been studied by using the Laplace transform. Numerical computation of analytical expression of deflection obtained after Inverse Laplace transform has been done using MATLAB software. The graphical observations have been discussed under various boundary conditions for different values of time and length. The above work has applications in design of resonators.

15

Modified Laplace based variational iteration method for the mechanical vibrations and its applications

Rehman Shahida, Hussain Akhtar, Rahman Jamshaid Ul, Anjum Naveed, Munir Taj

Acta Mechanica et Automatica

|

2022

|

Vol. 16, no. 2

98--102

EN

In this paper, we are putting forward the periodic solution of non-linear oscillators by means of variational iterative method (VIM) using Laplace transform. Here, we present a comparative study of the new technique based on Laplace transform and the previous tech-niques of maximum minimum approach (MMA) and amplitude frequency formulation (AFF) for the analytical results. For the non-linear oscillators, MMA, AFF and VIM by Laplace transform give the same analytical results. Comparison of analytical results of VIM by Laplace transform with numerical results by fourth-order Runge–Kutta (RK) method conforms the soundness of the method for solving non-linear oscillators as well as for the time and boundary conditions of the non-linear oscillators.

16

On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)

Sjödin Tord

Probability and Mathematical Statistics

|

2021

|

Vol. 41, Fasc. 1

1--7

EN

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y · X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

17

Asymptotic expansions for the first hitting times of bessel processes

Hamana Yuji, Kaikura Ryo, Shinozaki Kosuke

Opuscula Mathematica

|

2021

|

Vol. 41, no. 4

509--537

EN

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

18

Fourier, Laguerre, Laplace Transforms with applications

Aghili Arman

Journal of Mathematics and Applications

|

2021

|

Vol. 44

5--17

EN

In this article, the author considered certain time fractional equations using joint integral transforms. Transform method is a powerful tool for solving singular integral equations, integral equation with retarded argument, evaluation of certain integrals and solution of partial fractional differential equations. The obtained results reveal that the transform method is very convenient and effective. Illustrative examples are also provided.

19

Analysis of solutions of the 1D fractional Cattaneo heat transfer equation

Siedlecka Urszula, Ciesielski Mariusz

Journal of Applied Mathematics and Computational Mechanics

|

2021

|

Vol. 20, nr 4

87--98

EN

In this paper, a solution of the single-phase lag heat conduction problem is presented. The research concerns the generalized 1D Cattaneo equation in a whole-space domain, where a second order time derivative is replaced by the fractional Caputo derivative. The Fourier-Laplace transform technique is used to determine a solution of the considered problem. The numerical inversion of the Laplace transforms is applied. The effect of the order of the fractional derivative on the temperature distribution is investigated.

20

A method for Soft Fault Diagnosis of Linear Analog Circuits Using the Laplace Transform Technique

Tadeusiewicz Michał, Ossowski Marek, Korzybski Marek

International Journal of Electronics and Telecommunications

|

2021

|

Vol. 67, No. 3

531--536

EN

This paper is focused on multiple soft fault diagnosis of linear time-invariant analog circuits and brings a method that achieves all objectives of the fault diagnosis: detection, location, and identification. The method is based on a diagnostic test arranged in the transient state, which requires one node accessible for excitation and two nodes accessible for measurement. The circuit is specified by two transmittances which express the Laplace transform of the output voltages in terms of the Laplace transform of the input voltage. Each of these relationships is used to create an overdetermined system of nonlinear algebraic equations with the circuit parameters as the unknown variables. An iterative method is developed to solve these equations. Some virtual solutions can be eliminated comparing the results obtained using both transmittances. Three examples are provided where laboratory or numerical experiments reveal effectiveness of the proposed method.